# nLab Sandbox

$\array{ D \times S \times F & \simeq & (D \times H) \times_H (S \times F) & \overset{\simeq}{\longrightarrow} & (D \cdot H) \times_H (S \times F) & \longrightarrow & G \times_H ( S \times F ) \\ \big\downarrow && && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ D \times S &\simeq& (D \times H) \times_H S & \underoverset { m \times_H id } {\simeq} {\longrightarrow} & (D \cdot H) \times_H S &\hookrightarrow& G \times_H S }$

Throughout, let $G_1, G_2 \in$ TopologicalGroups and consider a continuous homomorphism of topological groups

$\phi \;\colon\; G_1 \longrightarrow G_2 \,.$

###### Definition

(pullback action)
Write

$Topological G_1 Spaces \overset{ \;\;\; \phi^\ast \;\;\; }{\longleftarrow} Topological G_2 Spaces$

for the functor which takes a topological G2-space $(X,\rho)$ to the same underlying topological space $\rho$, equipped with the $G_1$-action $\rho(\phi(-))$.

###### Lemma

The pullback action functor (Def. ) is the left adjoint of a pair of adjoint functors

$G_1 Spaces \underoverset { \underset{ Maps \big( G_2, - \big)^{G_1} }{\longrightarrow} } { \overset{ \phi^\ast }{ \longleftarrow } } {\bot} G_2 Spaces$

where for $X \in Topological G_1 Spaces$ the expression $Maps(G_2,-)^{G_1}$ denotes the $G_1$-fixed locus in the mapping space between topological spaces equipped with $G_1$-actions (on $G_2$ the $\phi$-induced left multiplication action) and equipped with the $G_2$-action given by

(1)$\array{ G_2 \times Maps(G_2,X)^{G_1} & \overset{}{\longrightarrow} & Maps(G_2,X)^{G_1} \\ (g_2, h) &\mapsto& h\big( (-) \cdot g_2 \big) \,. }$

###### Proof

To see the defining hom-isomorphism, consider a $G_1$-equivariant continuous function

$\phi^\ast X \overset{ \;\;\; f \;\;\; }{ \longrightarrow } Y \,.$

From this we obtain the following function

$\array{ X & \overset{ \tilde f }{ \longrightarrow } & Maps \big( G_2, Y \big)^{G_1} \\ x &\mapsto& \big( g_2 \mapsto f( g_2 \cdot x ) \big) \,, }$

where $e \in G_2$ denotes the neutral element.

This is manifestly:

• well-defined, due to the $G_1$-equivariance of $f$;

• continuous, being built from composition of continuous map;

• $G_2$-equivariant with respect to the action (1).

Conversely, given a $G_2$-equivariant continuous function $X \overset{\tilde f}{\longrightarrow} Maps\big(G_2, Y\big)^{G_1}$, we obtain the following function

$\array{ \phi^\ast X &\overset{}{\longrightarrow}& Y \\ x &\mapsto& \tilde f(x)(e) \,. }$

This is:

• continuous, being the composition of continuous functions;

• $G_1$-equivariant due to the equivariance properties of $\tilde f$:

\begin{aligned} \phi(g_1) \cdot x & \mapsto \tilde f \big( \phi(g_1)\cdot x \big) (e) \\ & = \tilde f ( x ) \big( e \cdot \phi(g_1) \big) \\ & = \tilde f ( x ) \big( \phi(g_1) \cdot e \big) \\ & = g_1 \cdot \big( \tilde f ( x ) ( e ) \big) \end{aligned}

Finally, it is clear that these transformations $f \leftrightarrow \tilde f$ are natural, hence it only remains to see that they are bijective:

Plugging in the above constructions we find indeed:

$\widetilde {\tilde f} \;\colon\; x \mapsto f(e \cdot x) = f(x)$

and

\begin{aligned} \widetilde {\widetilde {\tilde f}} \;\colon\; x & \mapsto \big( g_2 \mapsto \tilde f(g_2 \cdot x)(e) \big) \\ & = \big( g_2 \mapsto \tilde f(x)(e \cdot g_2) \big) \\ & = \big( g_2 \mapsto \tilde f(x)(g_2) \big) \,. \end{aligned}

$\array{ \widehat G \times_{\widehat H} S &\longrightarrow& \widehat G \times_{\widehat H} \widehat H &{\phantom{}}& \\ \big\downarrow && \big\downarrow \\ G \times_H S &\longrightarrow& G \times_H H }$
$\array{ \big( \Gamma \rtimes_\alpha G \big) \times_{\widehat H} S & \longrightarrow & \big( \Gamma \rtimes_\alpha G \big) / \widehat H \\ \big\downarrow && \big\downarrow \\ G \times_H S & \underoverset {}{}{\longrightarrow} & G/H }$

$\array{ \big( \Gamma \rtimes_\alpha G \big) \times_{\widehat H} S & \longrightarrow & \big( \Gamma \rtimes_\alpha G \big) \times_{\hat H} \hat H \\ \big\downarrow && \big\downarrow \\ G \times_{\hat H} S & \underoverset {}{}{\longrightarrow} & G \times_{\hat H} \widehat H }$

$\ldots$

$\infty$

Please do not delete the following example for the moment!

An article by vandenBergGarner2011 and another by MarraReggio2020.

## References

• Vincenzo Marra and Luca Reggio, A characterisation of the category of compact Hausdorff spaces, Theory Appl. Categ 35, 1871-1906 (2020) (arXiv:1808.09738)

• Benno van den Berg and Richard Garner, Types are weak $\omega$-groupoids, Proc. Lond. Math. Soc. (3) 102, No. 2, 370-394 (2011) (arXiv:0812.0298, doi:10.1112/plms/pdq026)

• Jiří Adámek and Jiří Rosický, How nice are free completions of categories?, Topology Appl. 273, 24 (2020)

FongSpivakTuyeras2017?

$\frac{13}{48} ( p_2 - \frac{1}{4} p_1^2 ) - \frac{1}{4} (\frac{1}{4} p_1^2)$
$\frac{13}{12} ( p_2 - \frac{1}{4} p_1^2 ) - (\frac{1}{4} p_1^2)$
$\frac{13}{12} p_2 - \frac{25}{12} (\frac{1}{4} p_1^2)$
$13 p_2 - 5 p_1^2$

$\Eta$

Write

$\array{ Q \\ \mathllap{ {}^{{}_{ h \mapsto (\phi(h),h) }} } \big\uparrow \\ H }$
$(\phi(g_1 g_2), g_1 g_2) = (\phi(g_1), g_1) \cdot (\phi(g_2), g_2) = \big( \phi(g_1) \cdot \alpha(g_1)(\phi(g_2)), g_1 \cdot g_2 \big)$
$\array{ (\gamma, u) &\mapsto& [\gamma, \sigma(u)] \\ \Gamma \times U & \underoverset{}{[id \times \sigma]}{ \rightleftarrows } & \big( \Gamma \rtimes_\alpha G_{\vert U} \big)/ Q \\ \gamma \phi(\sigma(u)), u && [\gamma, g] }$
$[\gamma,g] \;=\; (\gamma,g) \cdot ( \phi(g^{-1} \sigma(u)), g^{-1} \sigma(u) ) \;=\; ( \gamma \alpha(g)(\phi(g^{-1} \sigma(u))) )$
$\gamma \alpha(g)(\phi(g^{-1} \sigma(u))) = \gamma \alpha(g)( \phi(g^{-1}) \alpha(g^{-1})(\phi(\sigma(u))) ) = \gamma \alpha(g)(\phi(g^{-1})) \phi(\sigma(u))$
$\array{ G \times_H (S \times \Gamma) & \longrightarrow & G \times_H \Gamma & = & (\Gamma \times G) \big/ ( graph(H \to \Gamma) ) \\ \big\downarrow && \big\downarrow & \swarrow \\ G \times_H S & \underoverset {} {} {\longrightarrow} & G \times_H H }$

Last revised on April 8, 2021 at 10:46:35. See the history of this page for a list of all contributions to it.